The invention generally relates to arrangements and a methods for identifying parameters of a nonlinear model, which describes the nonlinear vibration of mechanical structures, such as used in electro-mechanical and electro-acoustical transducers. This information is the basis for identifying the constructional causes of the nonlinearities, linearizing the transfer behaviors of those transducers and for compensating actively nonlinear signal distortion in an electrical, mechanical or acoustical output signal.
Loudspeakers and other electro-acoustical transducers use diaphragms, panels, shells and other mechanical structures to generate vibration and sound. At low frequencies the transducer can be modelled by a network comprising lumped elements because the major part of the sound radiating surface vibrates as a rigid body and only the suspension (e.g. spider and surround in a loudspeaker) is deformed. This model can also consider nonlinearities inherent in the mechanical suspension and motor of the transducer and is the basis for the measurement and control applications as described in the publication by Yeh, D.T., Bank, B. Karjalainen, M. entitled “Nonlinear Modeling of a Guitar Loudspeaker Cabinet” in Proceedings of 11th Int. Conference on Digital Audio Effects, pp. DAFx1-DAFx-8, September 2008 and in the patent application US 2005/0031139. The patent application US 2003/0142832 uses the nonlinear lumped parameter model to develop a recursive structure.
At higher frequencies the mechanical structure generates higher-order vibration modes which require more complex modeling using distributed parameters. The publication by Yeh, D.T. and the patent application US 2005/0175193 use linear systems (e.g. equalizers) for the simulation of the higher-order modes and the active correction of the loudspeaker's transfer behaviors at small amplitudes. However, the relationship between forces and displacement becomes nonlinear at higher amplitudes and additional spectral components (harmonic and intermodulation distortion) are generated. Those distortions impair the quality of the sound reproduced by audio devices and the performance of active noise reduction and echo cancelation.
Nonlinear vibration and the sound radiation of higher-order modes can be described by analytical or numerical models (BEM, FEM) which require detailed information on the geometry and the material used in the mechanical components.
N. Queagebeur and A. Chaigne suggest in the publication “Mechanical Resonances and Geometrical Nonlinearities in Electrodynamic Loudspeakers”, Journal of Audio Eng. Soc., Vol. 56, No. 6 (2008), 462-471 the Karman model to describes the mechanical system on a higher abstraction level. This model requires the natural functions (modal shapes), natural frequencies and modal loss factor of the higher-order modes which can be determined by scanning the movement of the surface of the mechanical structure.
Generic black box models have been used for describing the nonlinear transfer behavior without considering the physical causes of the signal distortion. The document U.S. Pat. No. 6,687,235, for example, uses the Volterra expansion for echo compensation. The documents U.S. Pat. No. 5,148,427, U.S. Pat. No. 8,509,125, US2013/0216056, U.S. Pat. No. 6,813,311 and U.S. Pat. No. 5,329,586 use instead static nonlinearities without memory, which can be realized as tables, power series or nonlinear hardware components.